Abstract and Applied Analysis

Volume 2012 (2012), Article ID 195310, 11 pages

doi:10.1155/2012/195310

Research Article

The Solution of a Class of Third-Order Boundary Value Problems by the Reproducing Kernel Method

Zhiyuan Li,¹ Yulan Wang,¹ and Fugui Tan²

¹Department of Mathematics, Inner Mongolia University of Technology, Hohhot 010051, China

²Jining Teachers College, Jining 012000, China

Received 1 July 2012; Accepted 20 September 2012

Academic Editor: Wing-Sum Cheung

Copyright © 2012 Zhiyuan Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

This paper expands the application of reproducing kernel method to a class of third-order boundary value problems with mixed nonlinear boundary conditions. The analytical solution is represented in the form of series in the reproducing kernel space. The -term approximation is obtained and is proved to converge to the analytical solution. The numerical examples are given to demonstrate the computation efficiency of the presented method. Results obtained by the method indicate that the method is simple and effective.

1. Introduction

Third-order differential equations arise in a variety of different areas of applied mathematics and physics, for example, in the deflection of a curved beam having a constant or varying cross section, a three layer beam, electromagnetic waves, or gravity driven flows. Third-order boundary value problems were discussed in many papers in recent years, for instance, see [1–6] and references therein. In [1–3], the authors used the spline functions to solve boundary value problems. In [4], the authors developed a second-order method for solving third-order three-point boundary value problems based on Padé approximant in a recurrence relation. In [5], the authors introduced Adomian decomposition method for multipoint boundary value problems (BVPs). In this paper, we use reproducing kernel to solve singular third-order boundary value problems with mixed boundary conditions. Recently, the reproducing kernel methods [7–10] emerge one after another. Using the reproducing kernel methods, the authors discussed two-point boundary value problems and periodic boundary value problems. For third-order boundary value problems with mixed nonlinear boundary conditions, however, this method has not yet been applied. In previous work, the reproducing kernel method cannot be used directly to solve third-order boundary value problems with nonlinear boundary conditions. Our work is to present a numerical algorithm for solving a class of singular third-order boundary value problems. By using this method, the analytical solution and approximate solution are given and uniformly converge to the exact solution and its corresponding derivatives. The algorithms are efficiently applied to solve some model problems.

Let us consider the following singular problems of third-order ordinary differential equations:

where are known functions. are linear independence conditions of determining the solution. We assume that has a unique solution which belongs to , where which is a reproducing kernel space is defined in the second section.

In order to solve , let . It is easy to prove that is bounded linear operator.On the other hand, we assume that the conditions of determining the solution can be homogenized; after homogenization of these conditions, we put the conditions into the reproducing kernel space constructed in the following section. Equation can be transformed into the following form in :

To solve problem (1.2), we give a space . The inner product in is given by . Like in [8], we can get the following reproducing kernel space.

Theorem 1.1. The space is a reproducing kernel space and its reproducing kernel is , and

where

,  the symbol indicates that the operator applies to the function of .

2. The Reproducing Kernel Method

Let , Practise Gram-Schmidt orthonomalization for , we get

where are coefficients of Gram-Schmidt orthonormalization.

Theorem 2.1. If is distinct points dense in and is existent, then

is an analytical solution of the problem (1.2).

Proof. Since is an orthonormal systems, is expressed as

We denote the approximate solution of by

Theorem 2.2. Let where ,   are given by (2.2) and (2.4), then the sequence of number is monotone decreasing and .

Proof. Because

clearly and consequently is monotone decreasing in the sense of . By Theorem 2.1, we know is convergent in the norm of , then we have

Hence, .

Theorem 2.3 (convergence analysis). If , are given by (2.2) and (2.4), then and uniformly convergent to and , where .

Proof. For any

then there exists such that, .

Theorem 2.4. If is distinct points dense in and are given by (2.2) and (2.4), then .

Proof. We may set projective operator . Hence,

Theorem 2.5 (error estimate). If is distinct points dense in and , are given by (2.2) and (2.4), then , where .

Proof. For every given , there is always satisfying and . By Theorem 2.5 and implying , so we obtain

For application reproducing kernel property, we have

We also have

Moreover,

It is noted that we take norm of for variable . The function is derived on in the interval of , so we have . Hence,

3. Numerical Experiment

For showing the effectiveness of our method, we consider the following problems.

Example 3.1 (see [2, 3]). Considering the following third-order boundary values problem

where the exact solution is . By the homogeneous process of the boundary condition, let , problem can be transformed into the equivalent form

The numerical results are presented in Tables 1, 2, and 3.

Table 1: The absolute error of Example 3.1.

Absolute error

Present ()
Present ()
Present ()
Present ()
Ref. [3]
Ref. [2]
Present ()

0.0
0
0
0
0
0
0
0

0.1

0

0.3

0.4

0.5

0.7

0.9

1.0

0

0 s < CPU runtime ≤ 1 s
CPU runtime ≤ 10 s

CPU runtime = 180 s

Table 2: The numerical results of Example 3.1.

0.0
1
1
0
0
0.0000123873
0.0000123873

0.1
0.983602
0.983602

0.3
0.823414
0.823414

0.5

0.7

0.9

1.0

Table 3: The numerical results of Example 3.1.

0.0

0.010006

4.98999

0.2

0.0188419

0.4

0.0266921

0.6

0.0370827

0.8

0.050624

1.0

43.4925

Example 3.2 (see [11, 14]). Considering the following third-order obstacle problems:

where , the exact solution is

where

The numerical results are presented in Table 4.

Table 4: The observed maximum errors of Example 3.2.

Method

Our method

Nonpolynomial spline [11]

Quartic spline [12]

Finite difference [13]

Cubic spline [14]

Colloc. quantic spline [15]

Finite difference [16]

Quartic B spline [17]

Example 3.3. Considering the following boundary value problems with nonclassical condition:

We determine to get the true solution, given by . The numerical results are presented in Table 5.

Table 5: The numerical results of Example 3.3.

0.0
0
0
0

0.1
0.110517
0.110518

0.3
0.404958
0.404962

0.5
0.824361
0.824376

0.7
1.40963
1.40966

0.8
1.78043
1.78047

0.9
2.21364
2.21369

1.0
2.71828
2.71835

Example 3.4. Considering the following singular third-order three points boundary value problems with nonlinear condition

We determine to get the true solution, given by . The numerical results are presented in Table 6.

Table 6: The numerical results of Example 3.4, .

0.08

0.0273871
0.0273873

0.16

0.0490631
0.0490632

0.24

0.0655842
0.0655842

0.32

0.0773741
0.0773741

0.48

0.0878568
0.0878567

0.56

0.0868185
0.0868184

0.64

0.0815939
0.0815937

0.72

0.0720489
0.0720486

0.88

0.0389010
0.0389005

0.96

0.0144477
0.0144469

Example 3.5. Considering the following boundary value problems with nonlinear condition:

We determine to get the true solution, given by . The numerical results are presented in Table 7.

Table 7: The numerical results of Example 3.5.

0.08
0.000256
0.00025618

0.0096
0.009597

0.16
0.002048
0.0020481

0.0384
0.038399

0.24
0.006912
0.0069121

0.0864
0.0864

0.32
0.016384
0.0163841

0.1536
0.1536

0.4
0.032000
0.0320001

0.2400
0.2400

0.48
0.055296
0.0552961

0.3456
0.3456

0.56
0.087808
0.0878081

0.4704
0.4704

0.64
0.131072
0.131072

0.6144
0.6144

0.72
0.186624
0.186624

0.7776
0.7776

0.8
0.256000
0.256000

0.9600
0.9600

0.88
0.340736
0.340736

1.1616
1.1616

0.96
0.442368
0.442368

1.3824
1.3824

4. Conclusions and Remarks

In this work, we present an algorithm for solving third-order mixed boundary value problems (BVPs) based on the reproducing kernel method. The method can be generalized to get reproducing kernel of problem with linear conditions. All computations are performed by the Mathematica 7.0 software package.

Acknowledgments

This paper is supported by the Natural Science Foundation of Inner Mongolia (no. 2009MS0103) and the project of Inner Mongolia University of Technology (no. ZS201036).

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